Origins
The word 'organ' has two primary senses in modern English: a part of the body with a specific vital function (heart, liver, lungs) and a large musical instrument with pipes activated by keys. Both senses trace to a single Greek source, and their connection reveals how ancient thinkers conceived of function and purpose.
The word entered English in the thirteenth century from Old French 'organe,' from Latin 'organum,' from Greek 'organon' (ὄργανον). In Greek, 'organon' meant fundamentally 'a tool' or 'an instrument' — anything used to accomplish work. Aristotle titled his logical works the 'Organon' because he conceived of logic as the instrument of thought. Physicians applied the word to the parts of the body, viewing each organ as an instrument performing a designated task. Musicians applied it to instruments, and the pipe organ — the most elaborate mechanical instrument of antiquity — eventually claimed the word as its proper name.
The Greek word derives from the verb 'ergazesthai' (to work), from 'ergon' (work, deed), from PIE *werǵ- (to work, to do). This root is one of the great connecting threads of Indo-European vocabulary. Through Greek 'ergon,' it produced English 'energy' (en-ergeia, activity at work), 'ergonomic' (work-laws), 'synergy' (working together), 'metallurgy' (metal-working), 'surgery' (from 'cheir' + 'ergon,' hand-work), and 'liturgy' (public work, public service). Through Germanic, the same PIE root *werǵ- produced 'work' itself, making 'organ' and 'work' distant cousins.
Latin Roots
The derivative 'organic' originally meant 'serving as an instrument or organ' and 'relating to bodily organs.' Its modern meaning of 'derived from living organisms' developed in the eighteenth century when chemists distinguished 'organic' compounds (those found in living things) from 'inorganic' ones. The further extension to 'organic farming' (agriculture without synthetic chemicals) is a twentieth-century development.
'Organism' (a living being, viewed as an organized system of organs) was coined in the early eighteenth century. 'Organize' (to arrange into a functioning whole, as organs are arranged in a body) dates from the fifteenth century. 'Organelle' (a small organ-like structure within a cell) was coined by biologists in the early twentieth century.
The musical organ has a parallel history. The ancient Greek 'hydraulis' (water organ), invented in Alexandria in the third century BCE, was the first keyboard instrument. It used water pressure to maintain a steady air supply to pipes. The Latin 'organum' became the standard term for this instrument, and 'organum' in medieval music also referred to an early form of polyphony — singing in parallel intervals — because the technique mimicked the simultaneous sound of organ pipes.